Find the volume of the solid obtained by rotating about the y axis the region bounded by the curves.

Question

Quizplus · Accepted Answer

To find the volume of the solid, we need to use the formula for volume by rotation:

$V=\int_a^b\pi[f(y)]^2dy$

We can see from the graph that the two curves intersect at the point (1,1), so we will use this as our upper bound of integration. We also know that the two curves are symmetric, so we only need to find the volume of half the region, and then multiply by 2.

We will first find the equation of the curve labeled $y=x^3-3x+2$:

$x=y^{\frac13}-\frac{2}{y^{\frac13}}$

Then, we can set up our integral:

$V=2\int_0^1 \pi \left[\left(y^{\frac13}-\frac{2}{y^{\frac13}}\right)\right]^2 dy$

Simplifying, we get:

$V=2\int_0^1 \pi \left(y^{\frac{2}{3}}-\frac{4}{y^{\frac{1}{3}}}+4y^{\frac{2}{3}}\right) dy$

$V=2\pi \left[\frac{3}{5}y^{\frac{5}{3}}-\frac{8}{3}y^{\frac{2}{3}}+\frac{12}{5}y^{\frac{5}{3}}\right]_0^1$

$V=\frac{16}{5}\pi$

Therefore, the answer is B.

Find the Volume of the Solid Obtained by Rotating About